Quantum optics in phase space

Quantum Optics in Phase Space......Page 6
Contents......Page 14
1.1 On the Road to Quantum Optics......Page 24
1.2.1 Elastic Peak: Light as a Wave......Page 25
1.2.2 Mollow-Three-Peak Spectrum......Page 26
1.2.3 Anti-Bunching......Page 28
1.3.1 What is a Squeezed State?......Page 30
1.3.2 Squeezed States in the Optical Parametric Oscillator......Page 32
1.3.3 Oscillatory Photon Statistics......Page 35
1.3.4 Interference in Phase Space......Page 36
1.4 Jaynes-Cummings-Paul Model......Page 37
1.4.2 Time Scales......Page 38
1.5.1 An Amazing Maser......Page 39
1.5.2 Cavity QED in the Optical Domain......Page 42
1.6.1 Electron and Neutron Optics......Page 45
1.6.2 Atom Optics......Page 46
1.6.3 Atom Optics in Quantized Light Fields......Page 48
1.7.2 Quantum Information Processing......Page 49
1.8 Two-Photon Interferometry and More......Page 51
1.9 Outline of the Book......Page 52
2 Ante......Page 58
2.1.1 Properties of Eigenstates......Page 59
2.1.2 Derivative of Wave Function......Page 61
2.1.3 Fourier Transform Connects x- and p-Space......Page 62
2.2 Energy Eigenstate......Page 63
2.2.1 Arbitrary Representation......Page 64
2.2.2 Position Representation......Page 65
2.3.1 A State Vector is not Enough!......Page 67
2.3.2 Definition and Properties......Page 71
2.3.3 Trace of Operator......Page 72
2.3.4 Examples of a Density Operator......Page 74
2.4 Time Evolution of Quantum States......Page 76
2.4.1 Motion of a Wave Packet......Page 77
2.4.2 Time Evolution due to Interaction......Page 78
2.4.3 Time Dependent Hamiltonian......Page 80
2.4.4 Time Evolution of Density Operator......Page 84
3 Wigner Function......Page 90
3.1 Jump Start of the Wigner Function......Page 91
3.2.1 Marginals......Page 92
3.2.2 Overlap of Quantum States as Overlap in Phase Space......Page 94
3.2.3 Shape of Wigner Function......Page 95
3.3.1 von Neumann Equation in Phase Space......Page 97
3.3.2 Quantum Liouville Equation......Page 98
3.4.1 Definition of Moyal Function......Page 99
3.4.2 Phase Space Equations for Moyal Functions......Page 100
3.5 Phase Space Equations for Energy Eigenstates......Page 101
3.5.1 Power Expansion in Planck\'s Constant......Page 102
3.5.2 Model Differential Equation......Page 104
3.6 Harmonic Oscillator......Page 107
3.6.1 Wigner Function as Wave Function......Page 108
3.6.2 Phase Space Enforces Energy Quantization......Page 109
3.7 Evaluation of Quantum Mechanical Averages......Page 110
3.7.1 Operator Ordering......Page 111
3.7.2 Examples of Weyl-Wigner Ordering......Page 113
4 Quantum States in Phase Space......Page 122
4.1.1 Simple Phase Space Representation......Page 123
4.1.2 Large-m Limit......Page 124
4.1.3 Wigner Function......Page 128
4.2 Coherent State......Page 131
4.2.1 Definition of a Coherent State......Page 132
4.2.2 Energy Distribution......Page 133
4.2.3 Time Evolution......Page 136
4.3 Squeezed State......Page 142
4.3.1 Definition of a Squeezed State......Page 144
4.3.2 Energy Distribution: Exact Treatment......Page 148
4.3.3 Energy Distribution: Asymptotic Treatment......Page 151
4.3.4 Limit Towards Squeezed Vacuum......Page 155
4.3.5 Time Evolution......Page 158
4.4 Rotated Quadrature States......Page 159
4.4.1 Wigner Function of Position and Momentum States......Page 160
4.4.2 Position Wave Function of Rotated Quadrature States......Page 163
4.4.3 Wigner Function of Rotated Quadrature States......Page 165
4.5.1 Tomographic Cuts through Wigner Function......Page 166
4.5.2 Radon Transformation......Page 167
5.1 Probability for Classical Motion......Page 176
5.2 Probability Amplitudes for Quantum Motion......Page 178
5.2.1 An Educated Guess......Page 179
5.2.2 Range of Validity of WKB Wave Function......Page 181
5.3.1 Determining the Phase......Page 182
5.3.2 Bohr-Sommerfeld-Kramers Quantization......Page 184
5.4.1 Construction of Primitive WKB Wave Function......Page 186
5.4.2 Uniform Asymptotic Expansion......Page 187
6 WKB and Berry Phase......Page 194
6.1.1 Adiabatic Theorem......Page 195
6.1.2 Analysis of Geometrical Phase......Page 197
6.1.3 Geometrical Phase as a Flux in Hilbert Space......Page 198
6.2 WKB Wave Functions from Adiabaticity......Page 199
6.2.1 Energy Eigenvalue Problem as Propagation Problem......Page 200
6.2.2 Dynamical and Geometrical Phase......Page 204
6.2.3 WKB Waves Rederived......Page 206
6.3 Non-Adiabatic Berry Phase......Page 208
6.3.1 Derivation of the Aharonov-Anandan Phase......Page 209
6.3.2 Time Evolution in Harmonic Oscillator......Page 210
7.1 Outline of the Idea......Page 212
7.2.1 Jumps Viewed From Position Space......Page 215
7.2.2 Jumps Viewed From Phase Space......Page 220
7.3 Application to Franck-Condon Transitions......Page 223
7.4 Generalization......Page 224
8.1 Connection to Interference in Phase Space......Page 228
8.2 Energy Eigenstates......Page 229
8.3 Coherent State......Page 231
8.3.1 Elementary Approach......Page 232
8.3.2 Influence of Internal Structure......Page 235
8.4.1 Oscillations from Interference in Phase Space......Page 236
8.4.2 Giant Oscillations......Page 239
8.4.3 Summary......Page 241
8.5.1 Amplitude and Phase in a Classical Oscillator......Page 244
8.5.2 Definition of a Phase State......Page 246
8.5.3 Phase Distribution of a Quantum State......Page 250
9.1 What are Wave Packets?......Page 256
9.2 Fractional and Full Revivals......Page 257
9.3.1 Hierarchy of Time Scales......Page 260
9.3.2 Generic Signal......Page 262
9.4.1 The Early Stage of the Evolution......Page 264
9.4.2 Intermediate Times......Page 267
9.5.2 Shape Function......Page 269
10 Field Quantization......Page 278
10.1.1 Derivation of the Wave Equations......Page 279
10.1.2 Gauge Invariance of Electrodynamics......Page 280
10.1.3 Solution of the Wave Equation......Page 283
10.2.1 Solutions of Helmholtz Equation......Page 285
10.2.2 Polarization Vectors from Gauge Condition......Page 286
10.2.4 Boundary Conditions on the Magnetic Field......Page 287
10.2.5 . Orthonormality of Mode Functions......Page 288
10.3 The Field as a Set of Harmonic Oscillators......Page 289
10.3.1 Energy in the Resonator......Page 290
10.3.2 Quantization of the Radiation Field......Page 292
10.4.1 Zero-Point Energy of a Rectangular Resonator......Page 295
10.4.2 Zero-Point Energy of Free Space......Page 297
10.4.3 Difference of Two Infinite Energies......Page 298
10.4.4 Casimir Force: Theory and Experiment......Page 299
10.5.1 Vector Potential......Page 301
10.5.2 Electric Field Operator......Page 303
10.6 Number States of the Radiation Field......Page 304
10.6.2 Multi-Mode Case......Page 305
10.6.3 Superposition and Entangled States......Page 306
11.1 Properties of the Quantized Electric Field......Page 314
11.1.1 Photon Number States......Page 315
11.1.2 Electromagnetic Field Eigenstates......Page 316
11.2.1 Eigenvalue Equation......Page 318
11.2.2 Coherent State as a Displaced Vacuum......Page 320
11.2.3 Photon Statistics of a Coherent State......Page 321
11.2.4 Electric Field Distribution of a Coherent State......Page 322
11.2.5 Over–completeness of Coherent States......Page 324
11.2.6 Expansion into Coherent States......Page 326
11.2.7 Electric Field Expectation Values......Page 328
11.3.1 The Original Cat Paradox......Page 329
11.3.3 Wigner Phase Space Representation......Page 330
11.3.4 Photon Statistics......Page 333
12.1.1 Who Needs Phase Space Functions?......Page 344
12.1.2 Another Description of Phase Space......Page 345
12.2.2 Q-Functions of Specific Quantum States......Page 347
12.3.1 Heuristic Argument......Page 353
12.3.2 Rigorous Treatment......Page 356
12.4.1 Definition of P-Distribution......Page 360
12.4.2 Connection between Q- and P-Function......Page 361
12.4.3 P-Function from Q-Function......Page 362
12.4.4 Examples of P-Distributions......Page 364
13 Optical Interferometry......Page 372
13.1.1 Classical Treatment......Page 373
13.1.2 Symmetric Beam Splitter......Page 375
13.1.4 Transformation of Quantum States......Page 376
13.1.5 Count Statistics at the Exit Ports......Page 379
13.2.1 Classical Considerations......Page 380
13.2.2 Quantum Treatment......Page 381
13.3.1 Quantum State of the Output Modes......Page 384
13.3.2 Photon Count Statistics......Page 386
13.3.3 Simultaneous Measurement and EPR......Page 388
13.3.4 Q-Function Measurement......Page 390
13.4.1 Measurement of Classical Trigonometry......Page 393
13.4.2 Measurement of Quantum Trigonometry......Page 395
13.4.3 Two-Mode Phase Operators......Page 397
14 Atom-Field Interaction......Page 404
14.2 Vector Potential-Momentum Coupling......Page 405
14.2.1 Gauge Principle Determines Minimal Coupling......Page 406
14.2.2 Interaction of an Atom with a Field......Page 409
14.3.1 Expansion of Vector Potential......Page 412
14.3.3 Various Forms of the A - p Interaction......Page 413
14.3.4 Higher Order Corrections......Page 415
14.4.2 Röntgen Hamiltonians and Others......Page 416
14.5 Subsystems, Interaction and Entanglement......Page 418
14.6 Equivalence of A - p and r - E......Page 419
14.6.1 Classical Transformation of Lagrangian......Page 420
14.6.3 Matrix elements of A - p and r - E......Page 422
14.7 Equivalence of Hamiltonians H(1) and H(1)......Page 423
14.8.1 Derivation of the Hamiltonian......Page 425
14.8.2 Rotating-Wave Approximation......Page 429
15.1 Resonant Jaynes-Cummings-Paul Model......Page 436
15.1.1 Time Evolution Operator Using Operator Algebra......Page 437
15.1.2 Interpretation of Time Evolution Operator......Page 439
15.1.4 Dynamics Represented in State Space......Page 441
15.2.1 Atomic and Field States......Page 443
15.2.2 Rabi Equations......Page 445
15.3 Solution of Rabi Equations......Page 446
15.3.1 Laplace Transformation......Page 447
15.3.2 Inverse Laplace Transformation......Page 448
15.4 Discussion of Solution......Page 449
15.4.2 Resonant Case......Page 450
15.4.3 Far Off-Resonant Case......Page 452
16.1 Measurements on Entangled Systems......Page 458
16.1.1 How to Get Probabilities......Page 459
16.1.2 State of the Subsystem after a Measurement......Page 462
16.1.3 Experimental Setup......Page 463
16.2.1 Inversion as Tool for Measuring Internal Dynamics......Page 467
16.2.2 Experiments on Collapse and Revivals......Page 470
16.3.1 State Preparation with a Dispersive Interaction......Page 474
16.4.1 Outline of the Method......Page 477
16.4.2 Inverse Problem......Page 481
16.4.3 Example: Preparation of a Phase State......Page 484
17 Paul Trap......Page 496
17.1.1 No Static Trapping in Three Dimensions......Page 497
17.1.2 Dynamical Trapping......Page 498
17.2 Laser Cooling......Page 502
17.3 Motion of an Ion in a Paul Trap......Page 503
17.3.1 Reduction to Classical Problem......Page 504
17.3.2 Motion as a Sequence of Squeezing and Rotations......Page 506
17.3.3 Dynamics in Wigner Phase Space......Page 509
17.3.4 Floquet Solution......Page 513
17.4 Model Hamiltonian......Page 517
17.4.1 Transformation to Interaction Picture......Page 518
17.4.2 Lamb-Dicke Regime......Page 519
17.4.3 Multi-Phonon Jaynes-Cummings-Paul Model......Page 521
17.5 Effective Potential Approximation......Page 523
18 Damping and Amplification......Page 530
18.1 Damping and Amplification of a Cavity Field......Page 531
18.2.1 Coarse-Grained Equation of Motion......Page 532
18.3 Reservoir of Two-Level Atoms......Page 534
18.3.1 Approximate Treatment......Page 535
18.3.2 Density Operator in Number Representation......Page 537
18.3.3 Exact Master Equation......Page 542
18.4 One-Atom Maser......Page 545
18.4.1 Density Operator Equation......Page 546
18.4.2 Equation of Motion for the Photon Statistics......Page 547
18.4.3 Phase Diffusion......Page 552
18.5.1 Model and Equation of Motion......Page 555
18.5.2 First Order Contribution......Page 556
18.5.3 Bloch Equations......Page 558
18.5.4 Second Order Contribution......Page 560
18.5.5 Lamb Shift......Page 562
18.5.6 Weisskopf-Wigner Decay......Page 563
19.1.1 Dynamics......Page 572
19.1.2 Time Evolution of Probability Amplitudes......Page 575
19.2.1 Slowly Varying Approximation......Page 577
19.2.2 From Two Dimensions to One......Page 578
19.2.3 State Vector......Page 579
19.3.1 Heuristic Arguments......Page 580
19.3.2 Probability Amplitudes......Page 581
19.4.1 Measurement Schemes and Scattering Conditions......Page 582
19.4.2 Kapitza-Dirac Regime......Page 585
19.4.3 Kapitza-Dirac Scattering with a Mask......Page 591
19.5 Interference in Phase Space......Page 594
19.5.2 Area of Overlap......Page 595
19.5.3 Expression for Probability Amplitude......Page 596
20.1 Model......Page 602
20.2 Equation of Motion for Wigner Functions......Page 604
20.3 Motion in Phase Space......Page 605
20.3.2 Motion of the Atom in the Cavity......Page 606
20.3.3 Motion of the Atom outside the Cavity......Page 608
20.3.4 Snap Shots of the Wigner Function......Page 609
20.4.1 Distributions of Atoms in Space......Page 610
20.4.2 Focal Length and Deflection Angle......Page 612
20.5 Photon and Momentum Statistics......Page 613
20.6.1 Focal Length......Page 615
20.6.2 Focal Size......Page 617
A.1 Polynomial Ansatz......Page 620
A.2 Asymptotic Behavior......Page 622
A.2.2 Evaluation of the Integral Im......Page 623
A.2.4 Bohr\'s Correspondence Principle......Page 626
B.I Caution when Differentiating Operators......Page 628
B.2 Time Ordering......Page 629
B.2.1 Product of Two Terms......Page 630
B.2.2 Product of n Terms......Page 631
C.1 Why Other Measures Fail......Page 634
C.2 One Way out of the Problem......Page 635
C.3 Generalization to Higher Dimensions......Page 636
D.1 Formulation of the Problem......Page 638
D.2 Fourier Transform of Matrix Elements......Page 639
D.3 Kinetic Energy Terms......Page 640
D.4 Potential Energy Terms......Page 642
D.5 Summary......Page 643
E.1 Definition and Differential Equation......Page 644
E.2 Asymptotic Expansion......Page 645
E.2.1 Oscillatory Regime......Page 646
E.2.2 Decaying Regime......Page 647
E.2.3 Stokes Phenomenon......Page 648
F Radial Equation......Page 652
G Asymptotics of a Poissonian......Page 656
H.1.1 One-Dimensional Integrals......Page 658
H.1.2 Multi-Dimensional Integrals......Page 660
H.2 Cornu Spiral......Page 662
I.1 Diamond Transformed into a Rectangle......Page 666
I.2 Area of Diamond......Page 667
I.3 Area of Overlap as Probability......Page 669
J.1 Thermal State......Page 672
J.2 Photon Number State......Page 673
J.3 Squeezed State......Page 674
K.1 Explicit Evaluation of Kernel......Page 678
K.2 Strong Local Oscillator Limit......Page 679
L.1.1 Expansion of the Hamiltonian......Page 682
L.2 Classical Gauge Transformation......Page 684
L.2.1 Lagrangian with Center-of-Mass Motion......Page 685
L.2.3 Hamiltonian Including Center-of-Mass Motion......Page 686
L.3.1 Gauge Potential......Page 687
L.3.2 Schrödinger equation for f......Page 690
M Effective Hamiltonian......Page 692
N.1.1 Evaluation of Double Commutator......Page 694
N.2 Symmetry Relations in Trace......Page 696
N.2.2 Commutator Between Field Operators......Page 697
N.3 Master Equation......Page 698
N.4 Explicit Expressions for Γ, β and G......Page 699
N.5 Integration over Time......Page 700
O.1 Definition......Page 702
O.2 Asymptotic Expansion......Page 703
P Square Root of δ......Page 706
Q Further Reading......Page 708
Index......Page 711